*(that time I learnt the lesson: never promise what the next post will be about coz I often not deliver)*

Optimization algo need to explore the space of solutions, and this exploration is a major part of the algo! Let’s focus on one simple case: exploring in random order two finite dimensions. For example, let two integer variables, V1 taking value in interval [1, 100] and V2 taking value in interval [1, 20]: how to explore all the possible pairs (v1,v2) in random order?

A simple (almost-)solution is given by the following python code:

[(e1,e2) for e1 in shuffle(range([1,100)), for e2 in shuffle(range(1,20))]

which is in fact the cartesian product of the shuffles of V1 and V2 intervals … but this is ordered by values of the first shuffle and thus not random.

OK, now is the bad news: I have no solution completely space efficient to propose. My best effort is a solution which compute at least two shuffles, one on each dimension (kind of O(n + m) for space) and is even not random … just random enough for most of the cases π

So how? the solution is describe in this post http://weblog.raganwald.com/2007/02/haskell-ruby-and-infinity.html and especialy the section about the tabular view of the cartesian product. Did you read it? so the proposed solution is to navigate the cross-product table along its diagonal instead of row by row … simply smart isn’t it? It give “impression” of randomness π

Ocaml code for this algo is here: http://github.com/khigia/ocaml-anneal/tree/master/walks.ml (function pair_permutation_random_seq); It uses extensively an ad-hoc stream implementation (Seq module) to perform the walk lazyly. It was a good example to test the stream implementation!

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